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Theorem ifpdfan 35856
Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfan  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps , F.  ) )

Proof of Theorem ifpdfan
StepHypRef Expression
1 fal 1444 . . . 4  |-  -. F.
21intnan 922 . . 3  |-  -.  ( -.  ph  /\ F.  )
32biorfi 408 . 2  |-  ( (
ph  /\  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\ F.  ) )
)
4 df-ifp 1421 . 2  |-  (if- (
ph ,  ps , F.  )  <->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\ F.  )
) )
53, 4bitr4i 255 1  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps , F.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370  if-wif 1420   F. wfal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ifp 1421  df-tru 1440  df-fal 1443
This theorem is referenced by:  ifpdfnan  35877  ifpdfxor  35878
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